As known, selective catalytic reduction is a chemical process for reducing nitrogen oxides and mixtures thereof in the exhaust gases of internal combustion engines. In a selective catalytic reduction device, commonly called catalytic converter, a reducing chemical agent in liquid or gas state (in general ammonia or urea) is added to the exhaust gases. The reducer has the strong tendency to absorb oxygen, thus limiting the formation of nitrogen oxides inside the exhaust gas and forming H2O (steam) and N2 (nitrogen gas).
Currently, in SCR catalytic converters of motor vehicles a closed-loop Proportional-Integral-Derivative controller, commonly abbreviated as PID controller or regulator, is commonly used to maintain the pressure of the chemical agent within a predefined range of values. This allows the best operation both of the SCR catalyst, in particular in terms of the injection step of the chemical agent, and of the internal combustion engine.
In the SCR catalyst of a motor vehicle with an internal combustion engine the feeding unit of the chemical agent generally consists of a feeding pump provided with a filter. Using a PID controller, the speed of the driving motor of the feeding pump of the chemical agent is subordinated to a signal indicative of the instantaneous pressure value of such a chemical agent, which is continuously controlled and compared with the target value.
A generic PID controller acquires an input value from a process and compares it with a reference value. The difference between these two values, consisting of the so-called error signal, is thus used to determine the value of the output variable of the controller, which is the manipulable variable of the process. FIG. 1 shows a block diagram that illustrates the operation of a generic PID controller, in which:                r(t)=reference value;        e(t)=error signal;        u(t)=output variable of the controller;        d(t)=additive output disturbance of the controller;        y(t)=value to be controlled;        n(t)=measurement noise.        
In a PID controller the output variable u(t) is generated based on the contribution of three terms. The first term is proportional to the error signal e(t) between the reference value r(t) and the value to be controlled y(t). The second term is proportional to the integral of the error signal e(t), thus depending on the average value of the error signal e(t). The third term is, on the other hand, proportional to the derivative of the error signal e(t), thus being affected by the speed of variation of the error signal e(t). The control law of the PID controller can therefore be written as follows:
      u    ⁡          (      t      )        =                    K        P            ⁢              e        ⁡                  (          t          )                      +                  K        I            ⁢                        ∫          0          t                ⁢                              e            ⁡                          (              τ              )                                ⁢          d          ⁢                                          ⁢          τ                      +                  K        D            ⁢                          ⁢                        de          ⁡                      (            t            )                                    d          ⁢                                          ⁢          t                    where:                KP=proportional gain;        KI=integral gain;        KD=derivative gain.        
A motor vehicle with SCR catalyst must therefore be provided with a pressure sensor and with a PID controller on-board its electronic control station to allow the motor of the feeding pump of the chemical agent to close the pressure control loop of such a chemical agent. As mentioned previously, indeed, the speed of the driving motor of the feeding pump of the chemical agent is subordinate to a signal indicative of the instantaneous pressure value of such a chemical agent.
All of the external and disturbance factors, like for example temperature variation, viscosity and density values of the chemical agent, different internal tolerances of the pump, wearing of the pump, etc., are compensated by simply suitably adjusting the speed of the driving motor of the pump following the operative logic of a “black box” model. In the theory of systems, a “black box” model is a system essentially describable by its external behaviour, in other words only by how it reacts in output on the basis of a certain input stress, but the internal operation of which is not visible or is unknown.
Consequently, as known from the theory of controls and as shown in FIG. 2, a generic closed-loop PID controller carries out a feedback cycle that brings back, at the input of the process that it is wished to control, a function of the output that is added algebraically to the signal already present in input. For this reason, the closed-loop controllers are also called feedback controllers.